| Exam Board | OCR MEI |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Graphical optimization with objective line |
| Difficulty | Moderate -0.8 This is a straightforward linear programming question requiring translation of word constraints into inequalities and graphical representation. The constraints are clearly stated, the algebra is simple (0.10c + 0.05t ≤ 50, c + t ≥ 500, etc.), and part (i) essentially guides students through the interpretation. This is easier than average as it's a standard D1 textbook exercise with no novel problem-solving required. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06d Graphical solution: feasible region, two variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| At least 50% coffee (allow more than): number of coffee filters \(\geq\) number of tea bags, so number of tea bags \(\leq\) number of coffee filters. | B1 | referral to sales info to get \(\leq\) (allow \(<\)) |
| At most 75% coffee (allow less than): number of coffee filters \(\leq 3\times\) number of tea bags, so number of tea bags \(\geq \frac{1}{3}\times\) number of coffee filters. | B1 | referral to sales info + explanation of \(\frac{1}{3}\) to get \(\geq\) (allow \(>\)) |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Let \(x\) be the number of coffee filters. Let \(y\) be the number of tea bags (or vice versa). | B1 | "number of" essential |
| "500" line drawn | B1 | "500" line |
| £50 line drawn | B1 | £50 line |
| Lines from (i) drawn | B1 | lines from (i) |
| Correct shading | B1cao | shading |
| [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Coffee \(-\) 75% of 500. Tea \(-\) 50% of 500. | B1cao | |
| [1] |
# Question 3:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| At least 50% coffee (allow more than): number of coffee filters $\geq$ number of tea bags, so number of tea bags $\leq$ number of coffee filters. | B1 | referral to sales info to get $\leq$ (allow $<$) |
| At most 75% coffee (allow less than): number of coffee filters $\leq 3\times$ number of tea bags, so number of tea bags $\geq \frac{1}{3}\times$ number of coffee filters. | B1 | referral to sales info + explanation of $\frac{1}{3}$ to get $\geq$ (allow $>$) |
| **[2]** | | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Let $x$ be the number of coffee filters. Let $y$ be the number of tea bags (or vice versa). | B1 | "number of" essential |
| "500" line drawn | B1 | "500" line |
| £50 line drawn | B1 | £50 line |
| Lines from (i) drawn | B1 | lines from (i) |
| Correct shading | B1cao | shading |
| **[5]** | | |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Coffee $-$ 75% of 500. Tea $-$ 50% of 500. | B1cao | |
| **[1]** | | |
---3 Mary takes over a small café. She will sell two types of hot drink: tea and coffee.\\
A coffee filter costs her $\pounds 0.10$, and makes one cup of coffee. A tea bag costs her $\pounds 0.05$ and makes one cup of tea. She has a total weekly budget of $\pounds 50$ to spend on coffee filters and tea bags.
She anticipates selling at least 500 cups of hot drink per week. She estimates that between $50 \%$ and $75 \%$ of her sales of cups of hot drink will be for cups of coffee.
Mary needs help to decide how many coffee filters and how many tea bags to buy per week.\\
(i) Explain why the number of tea bags which she buys should be no more than the number of coffee filters, and why it should be no less than one third of the number of coffee filters.\\
(ii) Allocate appropriate variables, and draw a graph showing the feasible region for Mary's problem.
Mary's partner suggests that she buys 375 coffee filters and 250 tea bags.\\
(iii) How does this suggestion relate to the estimated demand for coffee and tea?
\hfill \mbox{\textit{OCR MEI D1 2015 Q3 [8]}}